   Chapter 4.2, Problem 22E

Chapter
Section
Textbook Problem

# Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. ∫ 1 4 ( x 2 − 4 x + 2 ) d x

To determine

To evaluate:

The integral 14x2-4x+2 dx by using the form of the definition of the integral given in theorem (4)

Explanation

1) Concept:

Theorem (4):

If f is integrable on [a, b] then

abfxdx=limni=1nfxi x

where x= b-an and xi=a+i x

f(x) is called as an integrand,n is the sub interval,a is the lower limit, and b is the upper limit.

2) Formula:

i)i=1ni2= nn+1(2n+1)6

ii)i=1ni= n(n+1)2

iii)i=1ncai=ci=1naiwhere c is a constant

iv)i=1n(ai-bi)=i=1nai-i=1nbi

v)i=1nc =nc

3) Given:

14x2-4x+2 dx

4) Calculation:

Here, a=1, b=4 and fx=x2-4x+2

Substituting value of a and b in x,

x= b-an

x= 4-1n

x= 3n

Now find xi,

xi=a+i x

xi=1+i 3n

xi=1+3in

By using theorem (4),

14x2-4x+2 dx=limni=1nf1+3in3n

= limn3ni=1nf1+3in

Thus by using formula (iii),

14x2-4x+2 dx= limn3ni=1nf1+3in

=limn3ni=1n1+3

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